thought, but we have no time to indulge in criticism. It suffices to point out, that, according to Kant himself, the existence of synthetic judgments a priori must seem utterly incredible to the unprejudiced mind. Nevertheless he believed that the actual existence of such judgments in science and in mathematics could not be denied; these disciplines seemed to be full of propositions — such as the principle of causality, and the Euclidean axioms — whose absolute validity could not seriously be doubted: so Kant believed himself confronted with the question: "These incredible propositions actually occur in the strictest of the sciences — how on earth are they possible?"
How is it to be explained, that we have knowledge, necessarily and absolutely valid knowledge — about facts of which we have not had any experience ? How can we be sure, that an event, which happens to-morrow or a hundred years from now, will have a cause ? or that seven objects and five objects, when counted on some distant, unknown star, will together be twelve objects ? — You know that the whole of the Critique of Pure Reason is devoted to the solution of this Problem. But alas ! there is no such problem, for there are no synthetic judgments a priori in natural science or in mathematics, or any where else. Kant must be excused for not recognizing the true nature of geometry, for in his time it was almost impossible to perceive that geometry, in so far as it dealt with the properties of space, is a physical science whose propositions are empirical, not a priori; and that in so far as it is a priori it is nothing but a hypothetical deductive system, consisting of propositional functions only, and consequently not asserting anything about any facts at all. Kant must perhaps also be pardoned for believing, even after Hume's criticism, in the absolute validity in the principle of causality, although the attitude of modern physics towards the principle proves that this belief is very far from being even a psychological necessity; but it is extremely difficult to justify Kant's opinion concerning the nature of arithmetical formular. His attempt to prove that 7+5 = 12 is a synthetic judgment seems very superficial and weak, especially if one considers Leibniz's treatment of the subject, and is one of the poorest passages in his whole work. Although there is at present still considerable disagreement about the ultimate foundation of mathematics nobady can nowadays hold the opinion any more that "arithmetical propositions" communicate any knowledge about the real world. They are certainly a priori, but their validity
